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With the advent of three-dimensional computer graphics, it has become attractive to many animators and scientists to create computer-generated animation. Such animations are used in the entertainment and advertising industries, for fine art, and increasingly for scientific visualization. The transformation of one shape to another, is generally known as metamorphosis. Traditional two-dimensional and three-dimensional animators have long exploited metamorphosis for its story telling possibilities. The technique of metamorphosis is powerful for conveying symbolic or actual correspondences between apparently different objects. Metamorphosis can also be used in computer aided design, for example, to combine a car body with a tear drop to form a more aerodynamic car body.
A problem with existing metamorphosis techniques is that they can produce unsatisfactory intermediate shapes unless user guidance is provided. However, once shapes become complicated, the amount of user guidance required can become extremely labor intensive.
Thus, it is desirable to provide an automatic process that transforms one shape to another while providing intermediate shapes meeting certain desiderata. For purposes of description, the initial object will be referred to as the "start shape" and the final object the "goal shape". The start and goal shapes together will be called the boundary shapes (or boundaries), and any shape produced along the way will be called a metamorph with respect to its boundaries. Any particular metamorph may be identified by a parameter .mu.: when .mu.=0 and .mu.=1, the metamorph is geometrically identical to the start and goal shapes respectively; intermediate values of .mu. generate intermediate (or proper) metamorphs.
It is desirable that the changes between shapes be geometrically, visually, and temporally smooth. Without these constraints one could simply show the start shape for a moment and then abruptly cut to the goal shape. Further criteria may be motivated by considering potential procedures, and deciding what features should be preserved or avoided. To start, consider a procedure which displays the start shape, shrinks it to a point by uniform scaling, and then runs the process in reverse, growing the goal shape from that point to its final form. The procedure is smooth, but degenerate. The nature of the degeneracy may be generalized by considering a complementary procedure, which enlarges the start shape to infinity in all directions, and then shrinks the infinite shape down to the goal shape. The infinite shape can be considered degenerate in the same sense as the point shape: all geometric information about the shape has been lost.
These two procedures demonstrate that one should avoid degeneracy. It is possible that a stronger size condition is desirable: metamorphs should grow no larger or smaller than the sizes of the boundary shapes. One example of a rule motivated by this condition is that the radius of the bounding sphere of each metamorph should always lie in the range defined by the radii of the bounding spheres of the boundary shapes.
Another criterion is illustrated by the following smooth, non-degenerate procedure. Assume for the moment that each boundary shape is made up of a collection of polygons or patches. First shrink each individual patch inwards towards its center, while simultaneously thickening it non-uniformly into a sphere. Once the shape has become a collection of spheres, some are split or joined until the number of spheres is equivalent to the number of patches in the goal shape. The process is then repeated in reverse, flattening each sphere in place until it forms a patch, which eventually all join together to form the goal shape. Like the previous procedure, this technique is not preferable for transforming shapes. The problem here is disconnection. The start and goal shapes are connected surfaces, and it is preferred that the metamorphs be connected in the same way. A stronger statement of this principle is that if the boundary shapes share the same topological genus, then all metamorphs should also be of that genus.
Another problem may be demonstrated by considering the transformation in FIG. 1. It shows two cones CS and CG as the start and goal shapes, respectively, and some metamorphs CM1, CM2, CM3, CM4 created by a possible procedure. In this example the apex of the start cone is moved to the apex of the goal cone, but not all the pieces of the cone move at the same speed. The metamorphs look odd, because they self-intersect, even though neither of the boundary shapes (CS, CG) does. This suggests another principle: If the start and goal shapes do not self-intersect, then neither should the metamorphs.
Thus, a metamorphosis should produce a temporally and geometrically continuous sequence of metamorphs that are non-degenerate, do not self-intersect, and are of constant genus if the boundary shapes are of the same genus.